Sutured manifolds and polynomial invariants from higher rank bundles

Abstract

For each integer N ě 2, Mariño and Moore defined generalized Donaldson invariants by the methods of quantum field theory, and made predictions about the values of these invariants. Subsequently, Kronheimer gave a rigorous definition of generalized Donaldson invariants using the moduli spaces of anti-self-dual connections on hermitian vector bundles of rank N . In this paper, Mariño and Moore's predictions are confirmed for simply connected elliptic surfaces without multiple fibers and certain surfaces of general type in the case that N " 3. The primary motivation is to study 3-manifold instanton Floer homologies which are defined by higher rank bundles. In particular, the computation of the generalized Donaldson invariants are exploited to define a Floer homology theory for sutured 3-manifolds. π 1 pΣ N pKqq with Σ N pKq being the N -fold cyclic branched cover of Y , branched along K. This verifies the Covering Conjecture, which asserts that Σ N pKq, for a non-trivial knot K, is not a homotopy sphere [50, Problem 3.38]. A consequence of the Covering Conjecture is the Smith Conjecture, stating that a non-trivial knot is not the fixed point set of an orientation preserving homeomorphism f : S 3 Ñ S 3 of order N [50, Problem 3.38]. The Covering Conjecture and the Smith Conjecture are both theorems, proved by geometrization techniques [1].Kronheimer and Mrowka's sutured Up2q-instanton homology group, SHI 2 , can be employed to answer Question 1.1 affirmatively for N " 2 (and hence for any even N ) and any non-trivial knot K [59]. 3 Associated to any knot K, there is a sutured manifold pM pKq, αpKqq where M pKq is the knot complement and αpKq is the union of two oppositely oriented meridional curves. Kronheimer and Mrowka proved that if the dimension of SHI 2 pM pKq, αpKqq is greater than 1, then there is a nonabelian representation of the knot group of K that satisfies (1.2). Similar to foliations, SHI 2 also behaves well with respect to surface decomposition, and one can inductively construct non-trivial elements of SHI 2 pM pKq, αpKqq after simplifying pM pKq, αpKqq by a series of sutured decomposition. In particular, the dimension of SHI 2 pM pKq, αpKqq is at least two for a non-trivial knot K. It is also shown in [31,7] that if K is a knot with non-trivial Alexander polynomial, then the answer to Question 1.1 is positive for infinitely many values of N . In the light of the success of SHI 2 in addressing Question 1.1, it is natural to look for the generalization of SHI 2 for higher values of N .

A Panorama Of Physical Mathematics c. 2022

“…In principle the formulae can be generalized to all higher rank compact semisimple gauge groups. This has not been done, and it would be interesting to do so since it would lead to nontrivial predictions about Floer homology following the line of reasoning in [968,969]. The mathematical challenges to verifying these physical predictions are formidable but nontrivial progress has been made [969,970].…”

Surgery, polygons and SU(N)‐Floer homology

“…Although [12,13] are mainly concerned with the Lie group U(2), the compactness theorems there can be adapted to the case of higher rank unitary Lie groups without any change. A concise review of the results that we use here is given in [11,Subsection 6.1]. See [11,12] for the definition of weak chain convergence.…”

“…A concise review of the results that we use here is given in [11,Subsection 6.1]. See [11,12] for the definition of weak chain convergence. a broken metric, a similar argument as in the previous paragraph shows that the connections A k , up to action of the gauge group and after passing to a subsequence, converge to an element of M p (X m , c; g ∞ ), that is, a broken ASD connection.…”

“…A concise review of the results that we use here is given in [DX17, Subsection 6.1]. See[Don02,DX17] for the definition of weak chain convergence.…”

Gauge theories on compact toric manifolds

“…It would be interesting to extend the present computations to gauge theories with fundamental and adjoint matter fields and perform a more thorough analysis of the higher rank cases. The latter point would provide new results for the Donaldson invariants in the higher rank case for which very few results are known at the moment, with the notable exception of [39,40]. Moreover, it would be useful for a large N analysis and for the study of holography for compact manifolds.…”